arXiv:cond-mat/9810107v1 [cond-mat.soft] 8 Oct 1998
Expansion of Liquid 4 He Through the Lambda
Transition
M.E. Dodd, P.C. Hendry, N.S. Lawson, P.V.E. McClintock and
C.D.H. Williams∗
Department of Physics, Lancaster University, LA1 4YB, UK
∗
School of Physics, University of Exeter, EX4 4QL, UK
Zurek suggested (Nature 317, 505; 1985) that the Kibble mechanism, through
which topological defects such as cosmic strings are believed to have been created in the early Universe, can also result in the formation of topological defects in liquid 4 He, i.e. quantised vortices, during rapid quenches through the
superfluid transition. Preliminary experiments (Hendry et al, Nature 368,
315; 1994) seemed to support this idea in that the quenches produced the predicted high vortex-densities. The present paper describes a new experiment
incorporating a redesigned expansion cell that minimises vortex creation arising from conventional hydrodynamic flow. The post-quench line-densities of
vorticity produced by the new cell are no more than 1010 m−2 , a value that
is at least two orders-of-magnitude less than the theoretical prediction. We
conclude that most of the vortices detected in the original experiment must
have been created through conventional flow processes.
PACS numbers: 11.27.+d, 05.70.Fh, 11.10.Wx, 67.40.Vs
1. INTRODUCTION
After a physical system has passed rapidly through a continuous phasetransition, its order-parameter can have components with large differences
between adjacent, but causally disconnected, “domains”. In such systems
topological defects can form at the domain boundaries.1 This idea was
proposed by Kibble2 in connection with the grand unified theory (GUT)
symmetry-breaking phase-transition of the early Universe, and has been developed by Zurek3, 4, 5 who has estimated how the density of defects created
M.E. Dodd et al.
depends on the rate at which the system passes through the transition. Zurek
also pointed out that this mechanism of defect production was applicable,
in principle, to all continuous phase-transitions, and that it should therefore be possible to validate some aspects of cosmological theories through
laboratory-scale experiments. The first of these examined weakly first-order
phase-transitions in liquid crystals.6, 7 Later, the corresponding experiments
were carried out using the second-order superfluid phase-transitions of liquid 4 He8 and liquid 3 He.9, 10 All these experiments produced defect densities
reported as being consistent with Zurek’s estimates.3, 4, 5
In this paper we describe an improved version of the 4 He experiment8
in which particular care has been taken to minimise the production of topological defects (i.e. quantised vortices) by ordinary hydrodynamic flow processes. Our new results, of which a preliminary report11 has already been
published, show no convincing evidence of any vortex creation at all. But
they allow us to place an approximate upper-bound on the initial density of
vortices produced by the Kibble-Zurek mechanism.
2. THEORETICAL BACKGROUND
A considerable amount is known about the properties12, 13, 14, 15 of vortices, and about how they are created at very low temperatures16, 17, 18 but
the mechanism responsible for the vorticity that appears19, 8 as a result
of passing through the λ-transition (which separates the normal helium-I
and the superfluid helium-II phases) is not understood. One possibility is
that pre-existing rotational flow caused by e.g. convection or boiling in the
helium-I phase is, with the onset of long-range order, converted into quantised vortex lines in helium-II; this is quite distinct from the Zurek scenario
which we shall now briefly describe.
The underlying idea3 is quite simple. A small isolated volume of
helium-I is initially held at pressure Pi , and temperature Ti , just above
the temperature Tλ (Pi ) of the λ-transition. The logarithmic infinity in its
heat capacity at Tλ makes it impossible to cool the sample quickly into the
superfluid phase but the pressure dependence of Tλ means that it can be
taken through the transition very rapidly by adiabatic expansion into the
helium-II phase, to a final pressure Pf and temperature Tf (Fig 1). Fluctuations present in the helium-I are expected to cause the nascent superfluid to
form with a spatially incoherent order-parameter, corresponding to a large
density of vortex lines. This scenario depends on the fact that the liquid can,
in principle, expand at velocities comparable to that of first-sound, whereas
the propagation velocity for changes in the order-parameter is limited by the
much slower velocity of second-sound.
Expansion of Liquid 4 He Through the Lambda Transition
The analogy3, 4, 5 between liquid helium and the early Universe arises
because they both be considered to undergo second-order phase transitions
describable in terms of Ginsburg-Landau theory.20 In each case the potential
contribution to the free-energy density can be written as:
1
(1)
V = α(T )|ψ|2 + β|ψ|4
2
where the parameter α is positive at temperatures above Tλ and negative below it, and β is a constant. For liquid 4 He the order-parameter is the modulus
of the complex-scalar field ψ, i.e. the Bose condensate wave-function which
is a solution of the Ginsburg-Pitaevskii equation.20 In the cosmological analogy the components of ψ are Higgs fields.4, 5 In the symmetric (helium-I)
phase T > Tλ and the time-average of the order-parameter hψi = 0. Below Tλ this gauge symmetry, is broken so hψi becomes non-zero, and the
real and imaginary parts of the potential in equation 1 acquire the same
“sombrero” shape as the corresponding cosmological free energy expressed
in terms of Higgs fields (Fig. 2). In the early Universe a symmetry-breaking
phase-transition from a false-vacuum state to a true-vacuum state is thought
to have occurred once the temperature had fallen to ∼ 1027 K, about 10−35 s
after the big bang. Although there are many variants of the basic model,
with and without inflation, it is believed that a variety of topological defects2
would have been produced in the transition because an event horizon prevented adjacent regions from being causally connected. Cosmic strings21 —
thin tubes of false vacuum — are one such defect and may have had a role
in galaxy formation. It is these that correspond to the quantised vortices
found in helium-II. The analogy between the helium and cosmological phase
transitions may thus be summarised as follows –
Higgs field 1
Higgs field 2
False vacuum
True vacuum
Cosmic string
←→
←→
←→
←→
←→
Re ψ
Im ψ
He I
He II
quantized vortex line
3. DETECTING VORTICITY
In our initial experiments,8, 23 the vortex density was measured by
recording the attenuation of a sequence of second-sound pulses propagated
through the helium-II. We expected that, following an expansion, the pulse
amplitude would grow towards its vortex-free value as the tangle decayed
and the attenuation decreased.
M.E. Dodd et al.
A considerable amount is known about the decay13, 14, 15, 24 of hydrodynamically created vortex tangles in helium-II. Numerical simulations24, 25
give a good qualitative description of the manner in which a homogeneous
isotropic tangle evolves and decays. The rate at which it occurs in this
temperature range is governed by the Vinen14 equation
h̄ 2
dL
= −χ2
L
dt
m4
(2)
where L is the vortex-line density at time t, the 4 He atomic mass is m4 , and
χ2 is a dimensionless parameter. The relationship between vortex density
and second-sound attenuation is known15 from experiments with rotating
helium, and may for present purposes be written
L=
6c2
ln(S0 /S)
κBd
(3)
where c2 is the velocity of second-sound, S and S0 are the signal amplitudes
with and without vortices present respectively. B is a weakly temperature
dependent parameter, κ = h/m4 is the quantum of circulation, and d is the
transducer separation. Integrating equation 2 and substituting for L from
equation 3 gives an expression for the recovery of the signal:
6c2
1
=
ln(S0 ) − ln(S)
κBd
1
χ2 κt
+
2π
L1
(4)
where L1 is the vortex density immediately after the expansion. All the
constants in this expression are known, although χ2 and B do not seem to
have been measured accurately within the temperature range of interest.
4. THE FIRST EXPERIMENT
A description of our first attempt to realise a bulk version4 of Zurek’s
experiment has been given in a previous paper23 but, briefly, the arrangement
was as follows. A cell with phosphor-bronze concertina walls was filled, by
condensing in isotopically26 pure 4 He through a capillary tube, and then
sealed with a needle-valve (Fig. 3). The top of the cell was fixed rigidly to
the cryostat but its bottom surface could be moved to compress the liquid, or
released to expand it, using a pull-rod from the top of the cryostat. The cell
was in vacuo surrounded by a reservoir of liquid 4 He at ∼ 2 K. A StratyAdams capacitance gauge27 recorded the pressure in the cell and carbonresistors were used as thermometers, one in the reservoir, one on the cell.
The temperature of the cell could be adjusted by means of an electrical heater
and a breakable thermal link to the 2 K reservoir. A trigger mechanism on
Expansion of Liquid 4 He Through the Lambda Transition
the mechanical linkage allowed the cell to increase its volume by ∼ 20% very
rapidly under the influence of its own internal pressure.
As described above, the vortex density was inferred from the amplitude
of second-sound pulses created with a thin-film heater. The signal was detected with a bolometer and passed, via a cryogenic FET preamplifier, to a
Nicolet 1280 data processor which recorded some ∼ 200 pulses during 1–2 s
following the expansion. The last pulses in the sequence define S0 , the signal
amplitude in the (virtual) absence of vortices.28 Fig. 4 shows examples of
data recorded with this first version of the experiment.
There was a “dead period” of about 50 ms after the mechanical shock of
the expansion which caused vibrations that obscured the signals. The initial
vortex-density was therefore obtained from plots such as Fig. 4(a) by backextrapolation to the moment t = 0 of traversing the transition and was found
to be ∼ 1012 − 1013 m−2 consistent with the theoretical expectation.3, 4, 5
However, an unexpected observation in these initial experiments8 was that
small densities of vortices were created even for expansions that occurred
wholly in the superfluid phase (Fig. 4c), provided that the starting point
was very close to Tλ . The phenomenon was initially23 attributed to vortices produced in thermal fluctuations within the critical regime, but it was
pointed out29 that effects of this kind are only to be expected for expansions
starting within a few microkelvin of the transition, i.e. much closer than the
typical experimental value of a few millikelvin. The most plausible interpretation — that the vortices in question were of conventional hydrodynamic
origin, arising from non-idealities in the design of the expansion chamber
— was disturbing, because expansions starting above Tλ traverse the same
region. Thus some, at least, of the vortices seen in expansions through the
transition were probably not attributable to the Zurek-Kibble mechanism
as had been assumed. It has been of particular importance, therefore, to
undertake a new experiment with as many as possible of the non-idealities
in the original design eliminated or minimised.
5. THE NEW EXPERIMENT
An ideal experiment would avoid all fluid flow parallel to surfaces during the expansion. This could, in principle, be accomplished by the radial
expansion of a spherical volume, or the axial expansion of a cylinder with
stretchy walls. In neither of these cases would there be any relative motion
of fluid and walls in the direction parallel to the walls, and hence no hydrodynamic production of vortices. However, it is impossible to eliminate the
effects of fluid-flow completely from any real apparatus. At the λ-transition
the critical velocity tends to zero, so any finite flow-velocity will create vor-
M.E. Dodd et al.
ticity. However, the period during which this happens is very short; the
entire expansion takes only a few milliseconds so only limited growth can
occur from the surface-vorticity sheet of half-vortex-rings.30
With hindsight, we can identify the principal causes of unintended vortex creation in the original experiment,8, 23 in order of importance, as follows:
(a) expansion of liquid into the cell from the filling capillary, which was closed
by a needle valve 0.5 m up-stream; (b) expansion of liquid out of a shorter
capillary connecting the cell to the Straty-Adams capacitive pressure-gauge;
(c) flow out of and past the fixed yoke (a U-shaped structure) on which the
second-sound transducers were mounted. In addition, (d) there were the
undesirable transients caused by the expansion system bouncing against the
mechanical stop at room temperature. The walls of the cell were made from
bronze bellows,8, 23 rather than being a stretchy cylinder, but the effects of
the flow parallel to the convoluted surfaces were relatively small.
The design of our new expansion cell (Fig. 5) addressed all the listed
problems, as follows: (a) a hydraulically-operated needle-valve eliminated
the dead-volume of capillary tube; (b) the phosphor-bronze diaphragm of the
pressure gauge27 became an integral part of the upper cell-wall eliminating
the long tube leading to the pressure gauge; (c) the cell was shortened from
25mm to ∼ 5mm so that the heater-bolometer pair could be mounted flush
with its top and bottom inner surfaces, thereby eliminating any need for a
yoke; (d) some damping of the expansion was provided by the addition of a
(light motor-vehicle) hydraulic shock-absorber.
The operation of the apparatus, the technique of data collection and
the analysis were much as described previously8, 23 except that the rate at
which the sample passed through the λ-transition was determined directly
by simultaneous measurements of the position31 of the pull-rod (giving the
volume of the cell, and hence its pressure) and the temperature of the cell.
The position detector made use of the variation of inductance of a
solenoid coil when a ferrite-core is moved in and out. The 1 cm long, 4
mm diameter, ferrite-core was rigidly attached to the top of the pull-rod,
where it emerged from the cryostat, and was positioned so that half of it
was inside the 5 mm bore of the solenoid. The solenoid itself was clamped to
the cryostat top-plate so that, as the cell expanded and the pull-rod moved
down, the ferrite penetrated further inside. The position was measured by
making the solenoid part of a resonant LCR circuit and setting the frequency
so that the output was at half maximum. The 2 MHz AC signal was rectified,
smoothed and then measured using a digital oscilloscope. The response time,
which was limited by the smoothing and the Q of the circuit, was ∼ 50µs
and the sensitivity was ∼500 mV/mm. This, and the linearity of the system,
were measured by calibration against a micrometer. The position could in
Expansion of Liquid 4 He Through the Lambda Transition
principle be measured to an accuracy of ∼ ±2µm but in the experiment this
became ∼ ±10µm because of the 8 bit digitiser of the oscilloscope.
We define the dimensionless distance from the transition by
ǫ=
Tλ − T
Tλ
(5)
dǫ
dt
(6)
and the quench time τq by
1
=
τq
ǫ=0
Fig. 6 shows a typical evolution of ǫ with time during an expansion. In
Fig. 6(a), which plots the full expansion period, it is evident that the system
“bounces” momentarily near ǫ = 0.02, but without passing back through the
transition. The effect is believed to be associated with the onset of damping
from the shock absorber, after backlash in the system has been taken up.
The quench time is readily determined from ǫ(t) near the transition. In
the case illustrated in Fig. 6, shown in expanded form in part (b), it was
τq = 17 ± 1 ms. We are thus able take both the pressure-dependence of Tλ ,
and the non-constant rate of expansion, explicitly into account.
6. RESULTS FROM THE NEW CELL
The improvements in the cell-design intended to eliminate vortices originating from conventional fluid-flow were clearly successful; as we had hoped,
and unlike the first cell, expansion trajectories that stay within the superfluid phase, even those starting as close as 42 mK to the transition, create
no detectable vorticity (Fig. 7).
We were surprised, however, to discover that no detectable vorticity was
created even when the expansion trajectory passed through the transition.
Signal amplitudes measured just after two such expansions are shown by
the data points of Fig. 8. It is immediately evident that, unlike the results
obtained from the original cell,8 there is now no evidence of any systematic
growth of the signal amplitude with time or, correspondingly, for the creation
of any vortices at the transition. One possible reason is that the density of
vortices created is smaller than the theoretical estimates,3, 4, 5 but we must
also consider the possibility that they are decaying faster than they can be
measured.
7. THE DECAY OF A VORTEX TANGLE
To try to clarify matters, we performed a subsidiary set of experiments,
deliberately creating vortices by conventional means and then following their
M.E. Dodd et al.
decay by measurements of the recovery of the second-sound signal amplitude.
By leaving the needle-valve open, so that ∼ 0.2 cm3 of liquid from the dead
volume outside the needle-valve actuator-bellows squirted into the cell during
an expansion, we could create large densities of vorticity and observe their
decay. Despite the highly non-equilibrium situation that arises immediately
following the expansion, as liquid squirts into the cell, it was found that the
temperature reached a steady value after ∼ 6 ms. Fig. 9 shows two examples
of attenuation plots ([ln(S/S0 )]−1 against time t) and these always, within
the experimental errors, had the linear form predicted by equation 4. Fig. 10
summarises the results of a number of expansions from which it was possible
to determine χ2 /B as a function of temperature and pressure. We found
that χ2 /B was weakly temperature-dependent and over the range of interest,
0.02 < ǫ < 0.06 it could be approximated by χ2 /B = 0.004 ± 0.001.
This measured value of χ2 /B was then used to calculate the evolution of
S/S0 with time for different values of L1 , assuming B = 1, yielding the
curves shown in Fig. 8. From the τq derived from the gradient in Fig 6, and
Zurek’s estimate (based on renormalisation-group theory) of
LRG =
L0
(τq /τ0 )2/3
where L0 = 1.2 × 1012 m−2 , τ0 = 100 ms
(7)
we are thus led to expect that L1 ≈ 4 × 1012 m2 . Direct comparison of the
calculated curves and measured data in Fig. 11 shows that this is plainly
not the case. In fact, the data suggest that L1 , the vorticity created by the
transition, is no more than 1010 m−2 , smaller than the expected value by at
least two orders-of-magnitude.
8. DISCUSSION
Given the apparently positive outcome of the earlier investigations,8
the null result of the present experiment has come as something of a surprise. There are several points to be made. First, Zurek did not expect his
estimates of L1 be accurate to better than one, or possibly two, orders-ofmagnitude, and his more recent estimate32 suggests somewhat lower defectdensities. So it remains possible that his picture3, 4, 5, 32 is correct for 4 He in
all essential details, and that an improved experiment with faster expansions
now being planned will reveal evidence of the Kibble-Zurek mechanism at
work in liquid 4 He. Secondly, it must be borne in mind that (2), and the
value of χ/B measured (from plots like those in Fig. 10) from the data of
Fig. 9, refer to hydrodynamically generated vortex lines. Vorticity generated
in the nonequilibrium phase transition might perhaps be significantly different, e.g. in respect of its loop-size distribution.33 It could therefore decay
Expansion of Liquid 4 He Through the Lambda Transition
faster, and might consequently be unobservable in the present experiments.
Thirdly, it is surprising to us that the 3 He experiments9, 10 seem to agree
with Zurek’s original estimates3, 4, 5 whereas the present experiment shows
that they overstate L1 by at least two orders-of-magnitude. It is not yet
known for sure why this should be, although an interesting explanation of
the apparent discrepancy has recently been suggested34 by Karra and Rivers.
They suggest that fluctuations near to Tλ may change the winding number,
i.e. reduce the density of vortices produced in the 4 He experiment, and they
show that the analogous density reduction would be much smaller in the
case of the 3 He experiments.
ACKNOWLEDGMENTS
We acknowledge valuable discussions or correspondence with S.N.
Fisher, A.J. Gill, R.A.M. Lee, R.J. Rivers, W.F. Vinen G.A. Williams and
W.H. Zurek. The work was supported by the Engineering and Physical Sciences Research Council (U.K.), the European Commission and the European
Science Foundation.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
A.J. Gill, Contemporary Phys. 39, 13 (1998).
T.W.B. Kibble, J. Phys. A 9, 1387 (1976).
W.H. Zurek, Nature 317, 505 (1985).
W.H. Zurek, Acta Physica Polonica B 24, 1301 (1993).
W.H. Zurek, Phys. Rep. 276, (1996).
I. Chuang, N. Turok and B. Yurke, Phys. Rev. Lett. 66, 2472 (1991);
I. Chuang, R. Durrer, N. Turok and B. Yurke, Science 251, 1336 (1991).
M.J. Bowick, L. Chander, E.A. Schiff and A.M. Srivastava, Science 263, 943
(1994).
P.C. Hendry, N.S. Lawson, R.A.M. Lee, P.V.E. McClintock and C.D.H.
Williams, Nature 368, 315 (1994).
C. Bäuerle, Y.M. Bunkov, S.N. Fisher, H. Godfrin and G.R. Pickett, Nature
382, 332 (1996).
M.H. Ruutu, V.B. Eltsov, A.J. Gill, T.W.B. Kibble, M. Krusius, Y.G. Makhlin,
B. Placais, G.E. Volovik and W. Xu, Nature 382, 334 (1996).
M.E. Dodd, P.C. Hendry, N.S. Lawson, P.V.E. McClintock and C.D.H.
Williams, “Non-appearance of vortices in fast mechanical expansions of liquid
4
He through the lambda transition”, preprint cond-mat/9808117, Phys. Rev.
Lett., in press.
H.E. Hall and W.F. Vinen, Proc. Roy. Soc. London A 238, 204 & 215 (1956).
W.F. Vinen, Proc. Roy. Soc. London A240, 114 & 128 (1957).
W.F. Vinen, Proc. R. Soc. Lond. A 242, 493 (1957).
M.E. Dodd et al.
15. R.J. Donnelly, Quantized Vortices in He II (Cambridge University Press, Cambridge, 1991).
16. E. Varoquaux, W. Zimmermann Jr. and O. Avenel, in Excitations in TwoDimensional and Three-Dimensional Quantum Fluids, ed. A.F.G. Wyatt and
H.J. Lauter, Plenum, New York, 1991.
17. C.M. Muirhead, W.F. Vinen and R.J. Donnelly, Phil. Trans. R. Soc. Lond. A
311, 433 (1984).
18. P.C. Hendry, N.S. Lawson, P.V.E. McClintock, C.D.H. Williams and R.M. Bowley, Phys. Rev. Lett. 60, 604 (1988); Phil. Trans. R. Soc. Lond. A 332, 387
(1990).
19. D.D. Awshalom and K.W. Schwarz, Phys. Rev. Lett. 52, 49 (1984).
20. V.L. Ginsburg and L.P. Pitaevskii, Soviet Phys. JETP 34, 858 (1958).
21. A. Vilenkin and E.P.S. Shellard, Cosmic Strings and Other Topological Defects
(Cambridge University Press, Cambridge, 1994).
22. A. Guth and P. Steinhardt, “The inflationary universe”, in The New Physics,
ed. P.C.W. Davies (Cambridge University Press, Cambridge, 1989).
23. P.C. Hendry, N.S. Lawson, R.A.M. Lee, P.V.E. McClintock and C.D.H.
Williams, J. Low Temp. Physics 93, 1059 (1993).
24. K.W. Schwarz and J.R Rozen, Phys. Rev. Lett. 66, 1898 (1991).
25. J. Koplik and H. Levine, Phys. Rev. Lett. 71, 1375 (1993).
26. P.C. Hendry and P.V.E. McClintock, Cryogenics 27, 131 (1987).
27. G.C. Straty and E.D. Adams, Rev. Sci. Instrum. 40, 1393 (1969).
28. M.E. Dodd, Ph.D. Thesis (University of Lancaster, 1998).
29. W.F. Vinen, Creation of Quantized Vortex Rings at the λ-Transition in Liquid
Helium-4, unpublished.
30. F.V. Kustmartsev, Phys. Rev. Lett. 76, 1880 (1996).
31. R.A.M. Lee, Ph.D. Thesis (University of Lancaster, 1994).
32. P. Laguna and W.H. Zurek, Phys. Rev. Lett. 78, 2519 (1997).
33. G.A. Williams, J. Low Temperature Phys. 93, 1079 (1993); and “Vortex loop
phase transitions in liquid helium, cosmic strings and high Tc superconductors”,
preprint cond-mat/9807338.
34. G. Karra and R.J. Rivers, “A re-examination of quenches in 4 He (and 3 He)”,
submitted to Phys. Rev. Lett. (1998).
35. P.C. Hendry, N.S. Lawson, R.A.M. Lee, P.V.E. McClintock and C.D.H.
Williams, Physica B 210, 209 (1995).
36. M.E. Dodd, P.C. Hendry, N.S. Lawson, R.A.M. Lee and P.V.E. McClintock,
Czech. J. Phys. 46, 43 (1996).
37. C.F. Barenghi, R.J. Donnelly and W.F. Vinen, J. Low Temp. Phys. 52, 189
(1983).
Expansion of Liquid 4 He Through the Lambda Transition
solid
Ti , Pi
P
lambda line
normal
liquid
superfluid
Tf , Pf
gas
T
Fig. 1. Schematic of expansion trajectory through the 4 He superfluid transition from a starting temperature and pressure (Ti , Pi ) to final values (Tf , Pf ).
M.E. Dodd et al.
False vacuum
True vacuum
Energy
Density
Higgs field 1
H i gg s
field 2
Fig. 2. Potential contribution to the free energy for the cosmological phase
transition, after Guth and Steinhardt22 .
Expansion of Liquid 4 He Through the Lambda Transition
heater
bolometer
Fig. 3. Schematic diagram showing the main features of the original experimental cell. The heater and bolometer were mounted on a yoke immersed
in the liquid 4 He.
M.E. Dodd et al.
1.0
S/Sref
0.8
0.6
0.4
0.2
0.0
(a)
0
200
400
600
800
1000
1200
t (ms)
1.2
1.0
S/Sref
0.8
0.6
0.4
(b)
0.2
0.0
0
200
400
600
800
1000
1200
t (ms)
1.0
0.8
S/Sref
0.6
0.4
0.2
0.0
(c)
0
200
400
600
800
1000
1200
t (ms)
Fig. 4. Recovery of the scaled second-sound signal amplitude S as a function
of time t in our original experiment, for various expansion trajectories: (a)
through the lambda line, Ti = 1.81 K, Pi = 29.6 bar, Tf = 2.05 K, Pf =
6.9 bar; (b) starting well below the lambda line, Ti = 1.58 K, Pi = 23.0 bar,
Tf = 1.74 K, Pf = 4.0 bar; (c) starting slightly (∼ 10 mK) below the lambda
line, Ti = 1.82 K, Pi = 25.7 bar, Tf = 2.03 K, Pf = 6.9 bar.
Expansion of Liquid 4 He Through the Lambda Transition
pressure gauge
diaphragm
phosphorbronze bellows
heater
4
He sample
orifice and needle
bolometer
sample entry port
pressurised liquid
4
He to close needle
valve
Fig. 5. The new expansion cell, designed so as to minimise hydrodynamic
creation of vortices.
M.E. Dodd et al.
0.06
ε=(1-T)/ Tλ
0.04
0.02
0.00
(a)
-0.02
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
t (s)
ε=(1-T)/ Tλ
0.02
0.01
0.00
(b)
-0.01
-400
-200
0
200
400
600
t (µs)
Fig. 6. Distance versus time t for a typical quench: (a) during the complete
expansion; (b) enlargement of region near ǫ = 0. The reciprocal of the
gradient at the transition gives the value of the ‘quench time’ parameter, τq .
Expansion of Liquid 4 He Through the Lambda Transition
1.6
1.4
1.2
S/S0
1.0
0.8
0.6
0.4
(a)
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
t (s)
1.6
1.4
1.2
S/S0
1.0
0.8
0.6
0.4
(b)
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
t (s)
1.6
1.4
1.2
S/S0
1.0
0.8
0.6
0.4
(c)
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
t (s)
Fig. 7. Second-sound pulse amplitude S/S0 as a function of time following
pressure quenches entirely within the helium-II phase: (a) starting far (∼
490 mK) below Tλ with Ti = 1.37 K, Pi = 24.2 bar, Tf = 1.47 K, Pf = 7.2 bar;
(b) starting ∼ 150 mK below Tλ with Ti = 1.74 K, Pi = 23.8 bar, Tf = 1.76 K,
Pf = 6.9 bar; (c) starting slightly (∼ 40 mK) below Tλ with Ti = 1.85 K,
Pi = 22.4 bar, Tf = 1.98 K, Pf = 6.4 bar.
M.E. Dodd et al.
1.2
1.0
S/S0
0.8
0.6
0.4
(a)
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
t (s)
1.4
1.2
S/S0
1.0
0.8
0.6
0.4
(b)
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
t (s)
1.2
1.0
S/S0
0.8
0.6
0.4
(c)
0.2
0.0
0
1
2
3
4
5
t (s)
Fig. 8. Second-sound pulse amplitude S/S0 as a function of time following pressure quenches through the λ transition: (a) starting close to the
transition, with Ti = 1.81 K, Pi = 30.3 bar, Tf = 2.03 K, Pf = 6.2 bar
(ǫi = −0.032); (b) starting further above the transition, with Ti = 2.05 K,
Pi = 22.7 bar, Tf = 2.09 K, Pf = 5.9 bar (ǫi = −0.089); (c) starting far above
the transition, with Ti = 1.96 K, Pi = 34.1 bar, Tf = 2.07 K, Pf = 6.1 bar
(ǫi = −0.167).
Expansion of Liquid 4 He Through the Lambda Transition
10
(a)
1/ln(S0/S)
8
6
4
2
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
t (s)
16
14
(b)
12
1/ln(S0/S)
10
8
6
4
2
0
0
100
200
300
400
500
600
t (ms)
Fig. 9. Examples of (lnS0 − lnS)−1 plotted versus time t for hydrodynamically created vortices.
M.E. Dodd et al.
0.007
0.006
0.005
χ2/B
0.004
0.003
0.002
0.001
0.000
-0.10
-0.08
-0.06
-0.04
-0.02
ε
Fig. 10. Measured values of the parameter χ2 /B as a function of distance ǫ
from the λ-transition.
Expansion of Liquid 4 He Through the Lambda Transition
1.0
S/S0
0.5
0.0
0.0
0.5
t (s)
Fig. 11. Evolution of the second-sound amplitude S with time, following an
expansion of the cell at t = 0 (data points), normalised by its vortex-free
value S0 . The open and closed symbols correspond to signal repetition rates
of 100 and 50 Hz respectively, and in each case the starting and finishing
conditions were ǫi = −0.032, ǫf = 0.039. The curves refer to calculated signal
evolutions for different initial line densities, from the bottom, of 1012 , 1011
and 1010 m−2 .
1.0